Certainly, let's focus on solving Problem 2 in a detailed, step-by-step manner:
### Problem 2
Given expression:
[tex]\[ \frac{x^2 + 12x - 45}{x^2 - 5x + 6} \div \frac{x^2 + 17x + 30}{x^4 - 16} \][/tex]
To simplify this expression, we follow the division of fractions procedure, which involves multiplying by the reciprocal. Let's break it down:
1. Rewrite the division as a multiplication by reciprocal:
[tex]\[
\frac{x^2 + 12x - 45}{x^2 - 5x + 6} \div \frac{x^2 + 17x + 30}{x^4 - 16} = \frac{x^2 + 12x - 45}{x^2 - 5x + 6} \times \frac{x^4 - 16}{x^2 + 17x + 30}
\][/tex]
2. Factorize where possible:
Factorize each polynomial if possible to simplify multiplication:
[tex]\[
x^2 + 12x - 45 = (x + 15)(x - 3)
\][/tex]
[tex]\[
x^2 - 5x + 6 = (x - 2)(x - 3)
\][/tex]
[tex]\[
x^2 + 17x + 30 = (x + 2)(x + 15)
\][/tex]
[tex]\[
x^4 - 16 = (x^2 - 4)(x^2 + 4) = (x - 2)(x + 2)(x^2 + 4)
\][/tex]
Substitute these factorizations back into the expressions:
[tex]\[
\frac{(x + 15)(x - 3)}{(x - 2)(x - 3)} \times \frac{(x - 2)(x + 2)(x^2 + 4)}{(x + 2)(x + 15)}
\][/tex]
3. Cancel common terms:
Cancel any common terms in the numerator and the denominator:
- Cancel out [tex]\((x - 3)\)[/tex] in the numerator and denominator of the first fraction.
- Cancel out [tex]\((x - 2)\)[/tex] in the numerator and denominator of the second fraction.
- Cancel out [tex]\((x + 2)\)[/tex] in the numerator and denominator.
- Cancel out [tex]\((x + 15)\)[/tex] in the numerator and denominator.
The expression simplifies to:
[tex]\[
\frac{1}{1} \times (x^2 + 4) = x^2 + 4
\][/tex]
So, the simplified answer is:
[tex]\[ x^2 + 4 \][/tex]
This matches the result you provided.
